# Preliminaries

• Tuğrul Dayar
Chapter
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)

## Abstract

The Kronecker (or tensor) product [50, 148] of two (rectangular) matrices A and B with A = [a(i A , j A )] is
$$A \otimes B = [a({i}_{A},{j}_{A})B].$$
Or, more formally, given $$A \in {\mathbb{R}}^{{n}_{A}\times {m}_{A}}$$ and $$B \in {\mathbb{R}}^{{n}_{B}\times {m}_{B}}$$, AB yields the (rectangular) matrix $$C \in {\mathbb{R}}^{{n}_{A}{n}_{B}\times {m}_{A}{m}_{B}}$$ whose entries satisfy
$$c({i}_{C},{j}_{C}) = a({i}_{A},{j}_{A})b({i}_{B},{j}_{B}),$$
$$\mbox{ with}\ \ \ {i}_{C} = {i}_{A}{n}_{B} + {i}_{B}\ \ \ \mbox{ and}\ \ \ {j}_{C} = {j}_{A}{m}_{B} + {j}_{B}$$
for
$$\begin{array}{rcl} ({i}_{A},{j}_{A}) \in \{ 0,\ldots,{n}_{A} - 1\} \times \{ 0,\ldots,{m}_{A} - 1\},& & \\ ({i}_{B},{j}_{B}) \in \{ 0,\ldots,{n}_{B} - 1\} \times \{ 0,\ldots,{m}_{B} - 1\},& & \\ \end{array}$$
where ×is the Cartesian product operator.

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